If you edit that link field in your content type (admin/structure/types/manage/your_contenttype/fields/field_c_button_link), there's a field Extra CSS-classes.

However, the classes entered here apply to all links created with 'field_c_buton_link'. If you want to add a class on one specific location, you might take a look at hook_preprocess_field] or even theme_link.

This is if you want the class on the <a> tag. The link class solution offered is easier, but when I tried it only applied to class to the wrapper of the a. So, if you're using Bootstrap for example, the link class module wouldn't work.

Thanks! This is very helpful, but presumes the field only has one item. If the field has multiple items you need to loop through them. e.g. if ($element['#field_name'] == 'field_link') { foreach ($variables['items'] as $key => $item){ $variables['items'][$key]['content']['#options']['attributes']['class'][] = 'blarg'; } }
– William MortadaDec 16 '16 at 12:36

It's easy to create your own formatter that overrides the link formatter. Although, now that I see there is a module for this (Link), you may want to use that one, as you can set it at the field level, rather than as a setting in the formatter. But I thought this might be useful to someone, who wants to build their own formatter for a link where you can add a class.